September 7th, 2009 Dr. Guido Grützner 1

Transcription

1 September 7th, 2009 Dr. Guido Grützner 1

2 Cautionary remarks about conclusions from the observation of record-life expectancy IAA Life Colloquium 2009 Guido Grützner München, September 7 th, 2009

3 Cautionary remarks about conclusions from the observation of record-life expectancy Introduction Simulation results Record-life expectancy as forecast-tool Conclusions September 7th, 2009 Dr. Guido Grützner 3

4 Introduction & some definitions Note: The LE definition of Oeppen&Vaupel is period LE at birth Definitions used in the talk Life expectancy (LE) is the expected number of further years of life We study the timedevelopment of specific LEs of nations or countries The record LE in a year is the maximal specific LE of all countries in scope LEs are seen as stochastic processes or time-series We study the slopes of the regression line of specific and record LE processes Source: Oeppen&Vaupel: Broken Limits to Life Expectancy, Science 10 May 2002:Vol no. 5570, pp September 7th, 2009 Dr. Guido Grützner 4

5 What to expect from this presentation Questions raised and (partially) answered - What can record LE tell us about LEs specific for a country? - What are drivers of record LE beyond specific LE improvements? - What issues need to be addressed before record LE can be used to forecast specific LEs? Tenor: - Record LE is a biased estimator of specific LEs - Inference on specific LEs needs to control all parameters driving the bias - Some of those parameters are only marginally relevant to LE development - To use record LE in forecasting you need a detailed model of the dependency-structure of global LE improvements September 7th, 2009 Dr. Guido Grützner 5

6 The basic issue in a nutshell Country A: current LE is 58 years - LE has equal chance of moving up or down 3 month next year - Expected improvement is zero or no systematic trend A: 58 1/2 1/2 Up: Down: Country B: same probabilities as A and independent B: 58 1/2 1/2 Up: Down: Record LE of A and B? - Probability of moving up: 3/4 - Probability of moving down: 1/4 Record LE is biased i.e. it overstates the expected LEs of A and B 58 A Up Down B Up Down Up Down Record LE = max(a,b) Up Up Up Down September 7th, 2009 Dr. Guido Grützner 6

7 Drivers of bias The simple example is already sufficient to demonstrate some key effects and drivers Number of countries: Increases bias - If three countries participate the probability of up is already 7/8 Simple correlation: Decreases bias - Can be demonstrated in the 2-stage tree: If outcome of B does depend on outcome of A Initial difference of LEs: Larger initial difference lowers the bias but it is still there - Needs multi-step consideration but is still straightforward September 7th, 2009 Dr. Guido Grützner 7

8 Cautionary remarks about conclusions from the observation of record-life expectancy Introduction Simulation results Record-life expectancy as forecast-tool Conclusions September 7th, 2009 Dr. Guido Grützner 8

9 The simulation laboratory Our goal is to demonstrate potential issues with record LE. So we strive for a simple model which is rich enough to show the effects. LE simulations are based on mortality rates from a Lee-Carter model 1) - Force of mortality: μ( x, t) = α x + β x κt - α xand β xfrom England&Wales population data, males, Stochastic time/series modelled as random walk with drift: κ t κt 1 = d + ε t with ε t iid, ε t ~ N(0, σ ) κ t Effects can be studied by systematic variation of parameters governing the stochastic properties - Drift, volatility, number of time series/countries, correlation-structure Reasonable range of variation is fixed by comparison to historic mortality data For practical reasons: LEs start with age 20 and are curtailed at 90. 1) see Lee-Carter (1992) or Cairns et.al. (2007) September 7th, 2009 Dr. Guido Grützner 9

10 Simulated LEs - first example Generate stochastic mortality rates Choose a number of countries (i.e. Time series) Calculate for each time t specific LEs and their record LE time series or countries 50 periods simulated Drift is zero Random walks are uncorrelated Record LE outlined in black Different time series make up the record LE line nodrift_regression.m September 7th, 2009 Dr. Guido Grützner 10

11 The record LE regression line Regress record LE over time to find the slope of record LE Time series same as before But regression lines are included Sampled is one concrete outcome Different samples will have different regression lines nodrift_regression.m September 7th, 2009 Dr. Guido Grützner 11

12 Improvements from nowhere Repeat sufficiently often to find the distribution of slopes of record LEs With sufficient countries record LE will show positive slope So the record LE shows improvement although all specific LEs have no drift 30 density of record LE s slope number of countries Mean slope 1-0.1% 2 1.5% 5 2.9% % % % density One country 50 countries 5 Nsim=5000 No rho, no drift, vola standard start = +-3% Main_sim.m slope of record LE increase September 7th, 2009 Dr. Guido Grützner 12

13 Potential amount of bias in record LE More realistic scenarios varying underlying drift and volatility There is indeed a possibility of material bias! kappa drift kappa volatility (of E&W) specific LE slope record LE slope overstatement due to bias 0 90% 0% 5% NA 0 100% 0% 5% NA 0 105% 0% 6% NA % 8% 12% 43% % 8% 12% 47% % 8% 12% 49% -1 90% 15% 17% 14% % 15% 17% 15% % 15% 17% 16% % 20% 20% 4% % 20% 20% 4% % 20% 20% 4% Comparison with data from the human mortality database 1) Assumptions on drift and volatility chosen to be reasonable in comparison to data Historic drift between -0.5 and -1.5 (E&W: -1.1) Standard deviation varies between 90% to 105% of E&W values Initial dispersion of starting positions +/- 3% of average LE 2003: 57years nsim = 5000, nperiod =50 rho = 3%, eps_init = +/-3% bias_estimate.m 1) see September 7th, 2009 Dr. Guido Grützner 13

14 How much bias in reality? The analysis presented is obviously not conclusive. So we do not know the true current or historic bias in record LE Some obvious missing points are: - Young and old ages are excluded - Only England & Wales data was used for Lee-Carter parameters - Static analysis i.e. parameters are fixed in advance - Extremely simple dependency: Multivariate normal random walk What is the role of the Lee-Carter model? - We do not claim that Lee-Carter is a particular good model for this - The claim is indeed: It doesn t matter which model you use - Lee-Carter is just a convenient way to generate stochastic mortality rates Remember the nutshell example: Volatility => Bias of record LE September 7th, 2009 Dr. Guido Grützner 14

15 Further examples and conclusion To decide on bias ALL parameters influencing the joint distribution need to be measured or at least their materiality estimated Example: Dynamic changes - Number of countries in scope: probably growing over time? - Changes in drift: catch-up to leading countries will lead to clustering, i.e. increased bias due to less differences between leaders - More complicated dependency, auto-regression of drift and error, changing volatility. As long as there is no evidence to the contrary it is safer to assume an unknown but potentially material bias in the slope of record LE. September 7th, 2009 Dr. Guido Grützner 15

16 Cautionary remarks about conclusions from the observation of record-life expectancy Introduction Simulation results Record-life expectancy as forecast-tool Conclusions September 7th, 2009 Dr. Guido Grützner 16

17 Record LE as forecasting-tool Record LE could be used to forecast specific LEs General idea - 1) Express any specific LE as a function of the record LE - 2) Forecast the record LE - 1) and 2) will give you immediately a forecast of your specific LE Example: specific LE = record LE - gap (Andreev, Vaupel 2006) LE record LE gap current specific LE forecast gap specific forecast Problem: - Bias in record LE is transferred to specific LE - Consistency is not ensured: If all countries had the same slope as record LE the record LE were different! now Time September 7th, 2009 Dr. Guido Grützner 17

18 Are other approaches viable? More complex approaches have been suggested - Lee (2006): Gap is not constant but decreases linearly over time - Torri (2008): Gap follows a stochastic process But any approach based on record LE is a potential victim of transfer of bias and lack of internal consistency Part of any proposed model should be a discussion of the consistency of assumptions between specific LE and measured/forecasted record LE - This most likely requires a description/analysis of the full dependency structure of all specific LEs involved Taking this complexity into account will limit the appeal of simplicity of the record LE approach. September 7th, 2009 Dr. Guido Grützner 18

19 Cautionary remarks about conclusions from the observation of record-life expectancy Introduction Simulation results Record-life expectancy as forecast-tool Conclusions September 7th, 2009 Dr. Guido Grützner 19

20 Conclusions Slope of record LE is a biased estimator of slope of any specific LE. - This is a mathematical property of the maximum of random variables and is not particular to human longevity or its temporal development. - Strength of bias is influenced by a potentially wide range of parameters. - Some of those parameters might only be marginally related to human longevity like e.g. the number of countries in scope. When forecasting based on the slope of record LE, - Care should be taken to prevent transfer of any bias from the record LE to the specific LE to be forecasted - Assumptions on the relationship between the forecasted specific LEs and their ensuing record LE should be checked for consistency Record LE is not a simple measure and analysing/controlling all influence factors will probably reduce its appeal of simplicity September 7th, 2009 Dr. Guido Grützner 20

21 Contact details You are welcome to send any questions, remarks and your opinion on the matter to You might also want to check out my company s website (our main focus is somewhat different though) September 7th, 2009 Dr. Guido Grützner 21

22 Cautionary remarks about conclusions from the observation of record-life expectancy Backup September 7th, 2009 Dr. Guido Grützner 22

23 Some sources and further reading Oeppen&Vaupel - Broken Limits to Life Expectancy, Science 10 May 2002:Vol no. 5570, pp Lee, R.D., and Carter, L.R. (1992) - Modeling and forecasting U.S. mortality", Journal of the American Statistical Association, 87: Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A., and Balevich, I.(2007) - A quantitative comparison of stochastic mortality models using data from England & Wales and the United States", Working paper, Heriot-Watt University, and Pensions Institute Discussion Paper PI Andreev, K. F. and J. W. Vaupel (2006). - Forecasts of Cohort Mortality after Age 50. Working paper, Max Planck Institute for Demographic Research, Rostock, Germany. Lee, R. (2006). - Mortality Forecasts and Linear Life Expectancy Trends In T. Bengtsson (Ed.), Prospectives on Mortality Forecasting. III. The Linear Rise in Life Expectancy: History and Prospects, pp Stockholm:National Social Insurance Board. Torri, Tiziana (2008) - Forecasting Life Expectancy in an International Context Paper presented to PPA September 7th, 2009 Dr. Guido Grützner 23

24 Development of some selected LEs 65 AUS FR 60 JAP NOR RUS LE (20-90) in years TAI SPA SWI ITA USA BEL BUL EST ICE IRE 40 LIT NTH NZ POL SWE Source of data: Human Mortality Database ( and own calculations September 7th, 2009 Dr. Guido Grützner 24

25 Parameters Parameters to vary and their implementation Parameter Number of countries Initial dispersion of LE Drift/Slope of LE Volatility of LE Correlation of LE Implementation Number of random mortality processes/les simulated Factor applied to the LE process Parameter of κ t distribution Parameter of κ t distribution Parameter of κ t distribution September 7th, 2009 Dr. Guido Grützner 25

26 Sample data used for comparison Male data SLOPE (from 1950) std deviation LE (2003) AUS 17% 8% 57.9 FR 15% 10% 55.7 JAP 23% 10% 57.8 NOR 7% 5% 56.9 RUS -15% 63% 39.7 TAI 17% 3% 54.5 SPA 14% 24% 56.1 SWI 15% 5% 57.7 ITA 14% 11% 57.1 USA 13% 4% 54.9 BEL 12% 7% 55.4 BUL -4% 42% 49.9 EST -5% 48% 46.7 ICE 12% 61% 59.0 IRE 11% 20% 55.8 LIT -10% 42% 46.8 NTH 6% 5% 56.3 NZ 12% 9% 57.1 POL 1% 19% 50.8 SWE 10% 4% 57.6 Avg 54.2 w/o negative 56.7 Source of data: Mortality tables: Human Mortality Database ( Derived values: own calculations September 7th, 2009 Dr. Guido Grützner 26

27 Lee Carter parameters alpha Based on ONS Data England and Wales male population Calibration on years Drift of kappa: -1.1 Standard deviation of ε: beta kappa main_mcalib.m September 7th, 2009 Dr. Guido Grützner 27